Suppose 3 grain boundaries intersect at a node as shown in image below. Derive the relationship between the mutual angles and the individual grain boundary tension (Show that\frac{\gamma_2_3}{sin\theta_1 }+\frac{\gamma_1_3}{sin\theta_2 }=\frac{\gamma_1_2}{sin\theta_3 }

and that the boundary tensions ?1, ?2 and ?3 must balance.

Under what conditions will the angles be equal?